CN111649066A - Control method of three-degree-of-freedom hybrid magnetic bearing - Google Patents
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
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- F16C—SHAFTS; FLEXIBLE SHAFTS; ELEMENTS OR CRANKSHAFT MECHANISMS; ROTARY BODIES OTHER THAN GEARING ELEMENTS; BEARINGS
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- F16C32/0444—Details of devices to control the actuation of the electromagnets
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- F16C—SHAFTS; FLEXIBLE SHAFTS; ELEMENTS OR CRANKSHAFT MECHANISMS; ROTARY BODIES OTHER THAN GEARING ELEMENTS; BEARINGS
- F16C32/00—Bearings not otherwise provided for
- F16C32/04—Bearings not otherwise provided for using magnetic or electric supporting means
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Abstract
The invention discloses a control method of a three-degree-of-freedom hybrid magnetic bearing, which is a closed-loop control system formed by sequentially connecting a controller implying a Lyapunov function flexible variable structure, a driving circuit, the three-degree-of-freedom hybrid magnetic bearing and a displacement detection module. The controller with the hidden Lyapunov function flexible variable structure is composed of a control parameter u, a system state vector x, a selection parameter p and a linear control vector k, and is connected in series with a driving circuit which directly controls the three-degree-of-freedom hybrid magnetic bearing. And detecting a rotor displacement signal of the three-degree-of-freedom hybrid magnetic bearing by using a displacement sensor, and feeding the rotor displacement signal back to the closed-loop controller to form a control system of the three-degree-of-freedom hybrid magnetic bearing with a Lyapunov function flexible variable structure. The invention has fast response speed, shortens the adjusting and settling time of the system, can effectively inhibit the influence caused by uncertainty and disturbance of the system and realizes the accurate and high-performance control of the three-degree-of-freedom hybrid magnetic bearing.
Description
Technical Field
The invention relates to a control method of a three-degree-of-freedom hybrid magnetic bearing, which is suitable for the precise control of a high-speed rotor magnetic suspension supporting system, provides conditions for the supporting of an ultra-high-speed rotor, and belongs to the field of high-speed and ultra-high-speed electric transmission.
Background
The magnetic bearing suspends the rotor in the air by utilizing electromagnetic force, so that no friction exists between the rotor and the bearing, and the magnetic bearing has the advantages of no friction, no abrasion, no need of lubricating oil, high supporting rotating speed, high rotor displacement precision, long service life and the like. The three-degree-of-freedom hybrid magnetic shaft is a highly nonlinear controlled system integrating electromechanical, electric and magnetic functions, is simple in mechanical structure, is easy to realize digital control, can reduce flywheel energy storage mechanical friction, and improves the critical rotating speed of a rotor.
The magnetic bearing system mainly comprises a magnetic bearing mechanical structure and a control system, wherein the mechanical structure influences the working performance of the whole magnetic bearing system, and the corresponding control system determines the implicit Lyapunov function performance, rigidity, damping and stability of the magnetic bearing system, so that the mechanical structure and the control system restrict whether a complete magnetic bearing system can realize the optimal working operation condition. With the rapid development of the industry, the three-degree-of-freedom hybrid magnetic bearing is unstable and complex to control due to the fact that factors such as critical rotating speed, load and interference of a rotor exist in an actual magnetic bearing control system, and the traditional PID control cannot well meet the requirements of the control system.
At present, the control research of the magnetic bearing mainly focuses on decoupling control and robust control, and common decoupling control methods include: the method comprises a differential geometry method, inverse system decoupling, neural network inverse decoupling, robust control and the like. The differential geometry method realizes the linearization of the nonlinear system by transforming the state of the nonlinear system and the input coordinate, but the mathematical tools used by the method are relatively hard and abstract, and the method is complex in calculation and is not beneficial to popularization. The inverse system theory is established based on a feedback linearization method, and a pseudo linear system is formed by integrating the inverse system and an original system, so that the control is performed by utilizing a mature linear system control theory, but the method excessively depends on an accurate mathematical model of the system, and the method is difficult to realize in the practical engineering application. The neural network inverse decoupling solves the problem that an accurate mathematical model of an inverse system is difficult to establish, but the neural network learning has the problems of low convergence speed and easy falling into a local minimum point. Although the sliding mode variable structure control is theoretically useful and strong in robustness, the sliding mode variable structure control is difficult to realize in practical engineering application.
The control method for the flexibility variable structure of the implicit Lyapunov function is improved by a nested Lyapunov function variable structure control method. The method utilizes the implicitly defined Lyapunov function as a selection strategy, so that the eigenvalue of the system continuously develops towards minus infinity, and rapid reaction is brought. Meanwhile, the infinite nested Lyapunov domain is utilized to improve the utilization of control constraint, and a time-lag observer is applied to observe unmodeled dynamics, uncertainty and external interference of the system and compensate, so that the robustness of the system to the uncertainty and the disturbance is improved.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a control system and a control method for a three-degree-of-freedom hybrid magnetic bearing with a built-in Lyapunov function flexible variable structure, so that the three-degree-of-freedom hybrid magnetic bearing system has good system response speed and control accuracy.
The three-degree-of-freedom hybrid magnetic bearing control method is a closed-loop control system formed by sequentially connecting a variable saturation flexible structure controller, a controlled object and a displacement detection module. The implicit Lyapunov function flexible variable structure controller is mainly determined by a control parameter u, a system state vector x, a selection parameter p and a linear control vector k. The controller u is u-kTv (x), wherein
g(v,x)＝0。
The technical scheme of the invention is as follows: a control method of a three-degree-of-freedom hybrid magnetic bearing is characterized in that an implicit Lyapunov function flexible variable structure controller, a drive circuit, a controlled object of the three-degree-of-freedom hybrid magnetic bearing and a displacement detection module are sequentially connected to form a closed-loop control system, the implicit Lyapunov function flexible variable structure controller is composed of a control parameter u, a system state vector x, a selection parameter p and a linear control vector, and the input of the implicit Lyapunov function flexible variable structure controller is a rotor reference position signal x given by a three-degree-of-freedom hybrid magnetic bearing rotor*、 y*、z*Displacement deviation e from rotor displacement output signals x, y, zx、ey、ezOutput as a force signal Fx *、Fy *，Fz *。
Further, the implicit Lyapunov function flexible variable structure controller may be expressed as
Wherein x is a state vector, A, B is a vector matrix, k is a vector matrix containing a parameter v, and g-0 is an output function;
according to the above equation, if the X-direction implicit Lyapunov function flexible variable structure controller a1 of the three-degree-of-freedom hybrid magnetic bearing is adopted, the state variable is taken
Wherein,
the system state equation can be expressed as:
wherein,
u＝ixm is rotor mass, kxFor the displacement stiffness, k, of the magnetic bearingixFor the current stiffness of the magnetic bearing ixTo output a current.
Further, the construction process of the implicit Lyapunov function flexible variable structure controller is as follows:
step 1: selecting a system matrix
Coefficient of characteristic polynomial of
Selecting
The best way is to select
Characteristic value λ ofi(1) Such a linear system
Will have good control performance, and in addition, choose to have any X ∈ X0Linear control vector of
Wherein X0Is to satisfy control constraints
Is determined, wherein | u ≦ a series of possible initial state vectors x (t ≦ 0) for | u ≦ u0For u control constraint, a is a constant vector;
step2 selection matrix R1The three constraints are
NR1+R1N＝-S1,
Wherein R is1、Q1Is a positive definite matrix, N ═ diag (-N., -1), e' (v) is the derivative of the error; there is usually a series of matrices that satisfy these conditions, from which matrix R is selected1So as to satisfy the domain G (1) { x | e (1) · xTR1x-1 < 0} is maximum; this domain is also the largest domain controlled by the compliance variable structure; due to the area G (1) of the ellipse and
proportional, optimal control problem solution
Obeying the three constraints;
wherein Q is1And S1Is an arbitrary positive definite matrix ifmin(Q1) And λmin(S1) The respective minimum eigenvalues are all positive and the error derivative e' (v) determines R1Then the optimal control problem has an analytical solution in this simple case, the optimal problem mentioned aboveThe solution to the problem is usually obtained using numerical methods such as log-barrier function:
the restricted problem is converted into a free problem:
wherein, the matrix R1Is composed of triangular matrix T, its transposed matrix and positive definite matrix R1In this case, the non-zero element of T may be any real number, the parameter r determines the depth of the threshold, and the free problem
Can be solved by a hill climbing method or an evolutionary algorithm;
step3 examination
If not, then restart at Step1, and in many cases, the three constraints are satisfied, which eliminates the need to solve the optimization problem and only needs to select the positive definite matrix Q1And
from the calculation of the matrix R1(ii) a Since N is a diagonal matrix, for R1Lyapunov equation NR1+R1N＝-S1Is usually satisfied, and furthermore, a third constraint
And in many cases is equally satisfactory.
Furthermore, the displacement detection module is formed by connecting a direction displacement sensor with a direction displacement interface circuit in series.
The control principle of the controller with the implicit Lyapunov function flexible variable structure is as follows:
(1) the implicit Lyapunov function flexible variable structure control is developed by nested Lyapunov function variable structure control. For the nested lyapunov function control method, every trace in the lyapunov domain will start from the domain and will not leave the domain. For a set of nested Lyapunov domains, it can limit the value of the control parameter and can guarantee maximum utilization of the control signal constraints in the regulation loop. In the regulation loop, the trajectory is limited to a gradually decreasing domain. As soon as a trajectory enters any domain, the controller associated therewith will function. Since the controller causes the response time of the control loop to decrease as p increases, the regulation will move towards a faster direction than if only a single controller were used. I.e. from one lyapunov domain into the next smaller domain, this ensures that sliding modes no longer occur.
(2) The nested Lyapunov function variable structure control method is discontinuous control, and on the basis, the implicit Lyapunov function flexible variable structure control is introduced. For a nonlinear system, firstly, the main part of the nonlinear system is linearized, and then an unmodeled dynamic state, uncertainty and external disturbance of the system are observed and compensated by directly using a time-lag observer, so that the linearized part of the nonlinear system can be designed by a linear system method.
The flexibility variable structure controller of the implicit Lyapunov function is input as a given rotor radial reference position signal x*，y*And z*Detecting the displacement deviation e of the rotor displacement output signals x, y and z of the three-freedom hybrid magnetic bearing with the displacement sensorx、eyAnd ezOutput as a force signal Fx *、Fy *，Fz *。
The invention has the advantages that:
1. by adopting the controller with the implicit Lyapunov function flexible variable structure, the high-precision tracking of the position can be realized, and simultaneously, the control signal can be continuous and smooth essentially, so the service life of the controller can be prolonged. Moreover, the adjusting and finishing time is shorter, and the system response is faster.
2. The time-lag observer is used as a closed-loop control method, unmodeled dynamics, uncertainty and external interference on the system are observed and compensated by using time lag, and the robust control method is excellent in performance and enables the system to have strong robustness.
Drawings
The invention is further illustrated with reference to the following figures and examples.
FIG. 1 is a block diagram of a three degree-of-freedom hybrid magnetic bearing control system;
FIG. 2 is a diagram of a controller with a flexible variant structure implying a Lyapunov function.
In the figure, a flexible variable structure controller of a Lyapunov function is hidden; a Lyapunov function flexibility variable structure controller is hidden in the x direction at a value of a1. x; a Lyapunov function flexibility variable structure controller is hidden in the a2.y direction; a flexible variable structure controller of the Lyapunov function is hidden in the z direction; b. a drive circuit; c. a displacement detection module; c1.z direction displacement sensor; c2.z direction displacement interface circuit; a c3.y direction displacement sensor; c4.y direction displacement interface circuit; c5.x direction displacement sensor; c6.x direction displacement interface circuit; d controlled object. u is a control parameter, f is a general operator, x is a system state vector, p is a selection parameter decision, k is a linear control vector, A is a matrix vector, B is a matrix vector,
is the first partial derivative of x.
Detailed Description
Referring to fig. 1, the control method of the three-degree-of-freedom hybrid magnetic bearing according to the present invention is a closed-loop control system formed by sequentially connecting an implicit lyapunov function flexible variable structure controller a, a controlled object d, a driving circuit b, and a displacement detection module c. The controller a with the implicit lyapunov function flexible variable structure can be specifically divided into an x-direction implicit lyapunov function flexible variable structure controller a 1; the y direction implies a Lyapunov function flexible variable structure controller a 2; square blockThe structure controller a3 is flexibly changed to the implicit lyapunov function. The input of the controller a1 with the implicit Lyapunov function flexible variable structure is a given three-degree-of-freedom hybrid magnetic bearing rotor reference position signal x*Deviation e from modulated rotor displacement output signal x rotor radial displacementxOutput as a force signal Fx *(ii) a The input of the controller a2 with the implicit Lyapunov function flexible variable structure is a given three-degree-of-freedom hybrid magnetic bearing rotor reference position signal y*Rotor radial displacement deviation e from modulated rotor displacement output signal yyOutput as a force signal Fy *(ii) a The input of the variable saturation flexibility variable structure controller a3 is a given three-degree-of-freedom hybrid magnetic bearing rotor reference position signal z*Rotor radial displacement deviation e from modulated rotor displacement output signal zzOutput as a force signal Fz *. The output signal controls the controlled object d through the driving circuit b.
The implicit Lyapunov function flexible variable structure control a of the invention is shown in FIG. 2 and can be expressed as
Wherein
R(v)＝D-1(v)R1D-1(v)
e (v) is a polynomial of order 2n or lower. Improved implicit Lyapunov function flexible variable structure control pair
Are all effective, i.e. for all
Are all flexibly variable structure control, wherein
Is the largest of all nested lyapunov domains. Without loss of generality, selection
The design steps are as follows:
taking the controller a1 with the flexibly-changed structure of the Lyapunov function hidden in the x direction as an example, the selection of the state variable is taken
Wherein x is the displacement of the magnetic bearing in the x direction. Combined rotor dynamic equation
Wherein k isx，ky，kzThe displacement rigidity coefficient of the magnetic bearing system at the balance position; k is a radical ofix，kiy，kizIs the current stiffness coefficient; m is the mass transfer, G is the rotor weight,
is the second derivative of x, ixFor controlling the current of the coil in the x-direction of the three-degree-of-freedom hybrid magnetic bearing, iyFor controlling the current of the coil in the y-direction of the three-degree-of-freedom hybrid magnetic bearing, izThe current of the coil is controlled in the z direction of the three-degree-of-freedom hybrid magnetic bearing.
The system state equation can be expressed as:
wherein,
u＝ix。
step 1: selecting a system matrix
Coefficient of characteristic polynomial of
Selecting
The best way is to select
Characteristic value λ ofi(1) Such a linear system
In addition, the choice is for any X ∈ X0Control vector of
Wherein X0Is to satisfy the control constraint | kTx|＝u0Is used to determine the series of possible initial state vectors x (t 0).
Step2 selection matrix R1The three constraints are
NR1+R1N＝-S1,
There are typically a series of matrices that satisfy these conditions. Selecting a matrix R from these matrices1So as to satisfy the domain G (1) { x | e (1) · xTR1x-1 < 0} is maximum. This domain is also the largest domain controlled by the compliance variable structure. Due to the area G (1) of the ellipse and
proportional, optimal control problem solution
Subject to the three constraints described above.
Wherein Q is1And S1Is an arbitrary positive definite matrix. If λmin(Q1) And λmin(S1) The respective minimum eigenvalues are all positive and e' (v) determines R1Then the optimal control problem has an analytical solution in this simple case. Solutions to the above-described optimization problem are typically obtained using numerical methods, such as the log-barrier function:
the restricted problem is converted into a free problem:
wherein, the matrix R1Is composed of a triangular matrix T and is the transpose thereof. And a positive definite matrix R1The non-zero elements of T may be any real number compared to the elements in this case. The parameter r determines the depth of the threshold. Free problem
Can be solved by a hill climbing method or an evolutionary algorithm.
Step3 examination
Whether or not it is satisfied. If not, then resume at Step 1. In many cases, the above three constraints are satisfied, which does not require solving the optimization problem, but only needs to select the positive definite matrix Q1And
from the calculation of the matrix R1. Since N is a diagonal matrix, for R1Lyapunov equation NR1+R1N＝-S1Is generally satisfactory. In addition, the third constraint
And in many cases is equally satisfactory.
And the constructed variable-saturation flexible structure controller a is connected in series in front of the drive circuit b and is used for controlling the drive circuit b, and the output of the drive circuit b is connected with the three-degree-of-freedom hybrid magnetic bearing d. In the displacement detection module c, an x-direction displacement sensor c5 and an x-direction displacement interface circuit c6 are sequentially connected, a y-direction displacement sensor c3 and a y-direction displacement interface circuit c4 are sequentially connected, and a z-direction displacement sensor c1 and a z-direction displacement interface circuit c2 are sequentially connected. The variable saturation flexible structure control is realized by software, and is written into a modularized program, so that the transplantation and application are facilitated. The driving circuit b and the displacement detection module c are realized by hardware.
In summary, the control method of the three-degree-of-freedom hybrid magnetic bearing of the present invention is a closed-loop control system formed by sequentially connecting a controller implying a Lyapunov function flexible variable structure, a driving circuit, the three-degree-of-freedom hybrid magnetic bearing, and a displacement detection module. The controller with the hidden Lyapunov function flexible variable structure is composed of a control parameter u, a system state vector x, a selection parameter p and a linear control vector k, and is connected in series with a driving circuit which directly controls the three-degree-of-freedom hybrid magnetic bearing. And detecting a rotor displacement signal of the three-degree-of-freedom hybrid magnetic bearing by using a displacement sensor, and feeding the rotor displacement signal back to the closed-loop controller to form a control system of the three-degree-of-freedom hybrid magnetic bearing with a Lyapunov function flexible variable structure. The invention adopts the controller with the implicit Lyapunov function flexible variable structure, can effectively eliminate the sliding mode, enables the system to be close to the optimal control system energy, and can obtain continuous and smooth control signals. The three-freedom-degree hybrid magnetic bearing has high response speed, shortens the adjusting and settling time of the system, can effectively inhibit the influence caused by uncertainty and disturbance of the system, and realizes accurate and high-performance control on the three-freedom-degree hybrid magnetic bearing.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
Claims (4)
1. A three-degree-of-freedom hybrid magnetic bearing control method is characterized in that: the closed-loop control system is formed by sequentially connecting an implicit Lyapunov function flexible variable structure controller, a driving circuit, a three-degree-of-freedom hybrid magnetic bearing controlled object and a displacement detection module, wherein the implicit Lyapunov function flexible variable structure controller is composed of a control parameter u, a system state vector x, a selection parameter p and a linear control vector, and the input of the implicit Lyapunov function flexible variable structure controller is a rotor reference position signal x given by a three-degree-of-freedom hybrid magnetic bearing rotor*、y*、z*Displacement deviation e from rotor displacement output signals x, y, zx、ey、ezOutput as a force signal Fx *、Fy *，Fz *。
2. The method for controlling a three-degree-of-freedom hybrid magnetic bearing according to claim 1, wherein:
the implicit Lyapunov function flexible variable structure controller can be expressed as
Wherein x is a state vector, A, B is a vector matrix, k is a vector matrix containing a parameter v, and g-0 is an output function;
according to the above equation, if the X-direction implicit Lyapunov function flexible variable structure controller a1 of the three-degree-of-freedom hybrid magnetic bearing is adopted, the state variable is taken
Wherein,
the system state equation can be expressed as:
wherein,
u＝ixm is rotor mass, kxFor the displacement stiffness, k, of the magnetic bearingixFor the current stiffness of the magnetic bearing ixTo output a current.
3. The method for controlling a three-degree-of-freedom hybrid magnetic bearing according to claim 2, wherein: the construction process of the implicit Lyapunov function flexible variable structure controller comprises the following steps:
step 1: selecting a system matrix
Coefficient of characteristic polynomial of
Selecting
The best way is to select
Characteristic value λ ofi(1) Such a linear system
Will have good control performance, and in addition, choose to have any X ∈ X0Linear control vector of
Wherein X0Is to satisfy the control constraint | kTx|＝u0Is determined, wherein | u ≦ a series of possible initial state vectors x (t ≦ 0) for | u ≦ u0For u control constraint, a is a constant vector;
step 2: selection matrix R1The three constraints are
NR1+R1N＝-S1，
Wherein R is1、Q1Is a positive definite matrix, N ═ diag (-N, …, -1), e' (v) is the derivative of the error; there is usually a series of matrices that satisfy these conditions, from which matrix R is selected1So as to satisfy the domain G (1) { x | e (1) · xTR1x-1 < 0} is maximum; this domain is also the largest domain controlled by the compliance variable structure; due to the area G (1) of the ellipse and
proportional, optimal control problem solution
Obeying the three constraints;
wherein Q is1And S1Is an arbitrary positive definite matrix ifmin(Q1) And λmin(S1) The respective minimum eigenvalues are all positive and the error derivative e' (v) is determinedR is fixed1The optimal control problem then has an analytical solution in this simple case, which is usually obtained using numerical methods, such as the log-barrier function:
the restricted problem is converted into a free problem:
wherein, the matrix R1Is composed of triangular matrix T, its transposed matrix and positive definite matrix R1In this case, the non-zero element of T may be any real number, the parameter r determines the depth of the threshold, and the free problem
Can be solved by a hill climbing method or an evolutionary algorithm;
step 3: examination of
If not, then restart at Step1, and in many cases, the three constraints are satisfied, which eliminates the need to solve the optimization problem and only needs to select the positive definite matrix Q1And
from the calculation of the matrix R1(ii) a Since N is a diagonal matrix, for R1Lyapunov equation NR1+R1N＝-S1Is usually satisfied, and furthermore, a third constraint
And in many cases is equally satisfactory.
4. The method for controlling a three-degree-of-freedom hybrid magnetic bearing according to claim 1, wherein: the displacement detection module is formed by connecting a direction displacement sensor with a direction displacement interface circuit in series.
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